Convergence of iterative methods based on Neumann series for composite materials: theory and practice

Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.Comment: 41 pages, 14 figures, 1 tabl

ON THE CONVERGENCE OF THREE ITERATIVE FFT-BASED METHODS FOR COMPUTING THE MECHANICAL RESPONSE OF COMPOSITE MATERIALS

International audienceThe last decade has witnessed a growing interest for the so-called " FFT-based methods " for computing the overall and local properties of heterogeneous materials submitted to mechanical solicitations. Since the original method was introduced by Moulinec and Suquet [1], several authors have proposed different algorithms to better deal with non-linear materials or with materials whith highly contrasted mechanical properties between their constituents. The present paper aims to compare three methods of this family of algorithms which were designed to accelerate the convergence of the scheme. The study concerns a linear elastic material-although the methods involved can be extended into the case of non-linear behavior-submitted to a prescribed overall strain E. The stiffness tensor c(x) of the material varies with the position x. The numerical method proposed by Moulinec & Suquet lies on the iterative resolution of the Lippmann-Schwinger equation and can be summarized by the following relation between two successive iterates ε i and ε i+1 of the strain field: ε i+1 (x) = −Γ 0 * (c(x) − c 0) : ε i (x) + E (1

Impact and Optimum Placement of Off-Shore Energy Generating Platforms

Understanding the influence of wave distribution, hydrodynamics and sediment transport is crucial for the placement of o -shore energy generating platforms. The TELEMAC suite is used for this purpose, and the performance of the triple coupling between TOMAWAC for wave propagation, TELEMAC-3D for hydrodynamics and SISYPHE for sediment transport is investigated for several mesh sizes, the largest grid having over 10 million elements. The coupling has been tested up to 3,072 processors and good performance is in general observed

Comparaison de 3 méthodes à base de transformées de Fourier pour le calcul des propriétés mécaniques de matériaux hétérogènes

International audienceLes simulations numériques du comportement mécanique des matériaux hétérogènes doivent tenir compte de la complexité de leurs microstructures. Au cours des deux dernières décennies, les méth-odes dites " à base de transformées de Fourier " ont retenu l'attention par leur efficacité et la simplicité de leur mise en oeuvre. On se propose dans cette étude de comparer 3 méthodes numériques de cette famile, proposant une convergence accélérée en comparaison de celle du schéma itératif proposé par Moulinec & Suquet ([1], [2]). On s'intéressera tout particulièrement aux conditions de convergence et au choix des paramètres permettant une convergence optimale de ces schémas. Mots clés — transformée de Fourier, matériaux hétérogène

Comparison of different FFT-based methods for computing the mechanical response of heteregoneous materials

International audienceThe last decade has witnessed a growing interest for the so-called " FFT-based methods " for computing the overall and local properties of heterogeneous materials submitted to mechanical solicita-tions. Since the original method was introduced by Moulinec and Suquet [1], several authors have proposed different algorithms to better deal with non-linear materials or with materials with highly contrasted mechanical properties between their constituents. The study concerns a linear elastic material-although the methods involved can be extended into the case of non-linear behavior-submitted to a prescribed overall strain E. The stiffness tensor c(x) of the material varies with the position x. The numerical method proposed by Moulinec & Suquet lies on the iterative resolution of the Lippmann-Schwinger equation and can be summarized by the following relation between two successive iterates ε i and ε i+1 of the strain field: ε i+1 (x) = −Γ 0 * (c(x) − c 0) : ε i (x) + E , where c 0 is the stiffness tensor of a reference medium supposed to be linear elastic, where Γ 0 is a Green operator associated to c 0 and where * denotes the convolution operator. Eyre & Milton [2], Michel et al. [3] and Monchiet & Bonnet [4] proposed different schemes to accelerate the convergence of the initial scheme. It has been recently demonstrated in [5] that the two first schemes are particular cases of the last one. On the other hand, Zeman et al. [6] proposed to use a conjugate gradient method for solving the Lippmann-Schwinger equation. The present paper aims to compare these different methods with a special attention paid to their relative efficiency and their rates of convergence

Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials

International audienceSince Moulinec & Suquet (1994, 1998) introduced an iterative method based on Fourier transforms to compute the mechanical properties of heterogeneous materials, improved algorithms have been proposed to increase the convergence rate of the scheme. This paper is devoted to the comparison of the accelerated schemes proposed by Eyre & Milton (1999), by Michel et al (2000) and by Monchiet & Bonnet (2012). It shows that the algorithms by Eyre-Milton and by Michel et al are particular cases of Monchiet-Bonnet algorithm, corresponding to particular choices of parameters of the method. An upper bound of the spectral radius of the schemes is determined, which enables to propose sufficient conditions of convergence of the schemes. Conditions are found for minimizing this upper bound. This study shows that the scheme which minimizes this upper bound is the scheme of Eyre & Milton. The paper discusses the choice of the convergence test used in the schemes

Wall-Adapting Local Eddy-Viscosity Turbulence Model for TELEMAC3D

Water Qualit

Energy-based drag decomposition analyses for a turbulent channel flow developing over convergent–divergent riblets

Direct numerical simulations of a turbulent channel flow developing over convergent-divergent (C-D) riblets are performed at a Reynolds number of Re-b = 2800, based on the half channel height delta and the bulk velocity. To gain an in-depth understanding of the origin of the drag generated by C-D riblets, a drag decomposition method is derived from kinetic energy principle for a turbulent channel flow with wall roughness. C-D riblets with a wavelength, lambda, ranging from 0.25 delta to 1.5 delta, are examined to understand the influence of secondary flow motions on the drag. It is found that as lambda increases, the intensity of the secondary flow motion increases first and then decreases, peaking at lambda / delta = 1. At lambda / delta & GE; 1, some heterogeneity appears in the spanwise direction for the turbulent kinetic energy (TKE) and vortical structures, with the strongest enhancement occurring around regions of upwelling. All the riblet cases examined here exhibit an increased drag compared to the smooth wall case. From the energy dissipation/production point of view, such a drag increase is dominated by the TKE production and the viscous dissipation wake component. While the drag contribution from the TKE production shear component decreases as lambda increases, the drag contribution from the wake component of both the TKE production and viscous dissipation follows the same trend as the intensity of the secondary flow motion. From the work point of view, the drag increase in the riblet case at lambda / delta = 0.25 comes mainly from the work of the Reynolds shear stresses, whereas at lambda / delta & GE; 1, the drag augmentation is dominated by the work of the dispersive stresses. At lambda / delta = 0.5, both components play an important role in the increase in the drag, which also exhibits a peak